-x
=e
INPUT: clc; funcprot(0) function ydot=f(x, y) ydot= exp(-x) endfunction x=0:0.1:5; y0=0; x0=0; y = ode(y0,x0,x,f); disp(y); plot2d(x,y)
OUTPUT: 0 0.0951626 0.1812692 0.2591818 0.3296799 0.3934693 0.4511883 0.5034147 0.550671 0.5934303 0.6321206 0.6671289 0.6988058 0.7274682 0.7534031 0.7768699 0.7981035 0.8173165 0.8347011 0.8504314 0.8646647 0.8775436 0.8891968 0.8997411 0.909282 0.917915 0.9257264 0.9327945 0.9391899 0.9449768 0.9502129 0.9549508 0.9592378 0.9631168 0.9666267 0.9698026 0.9726762 0.9752764 0.9776292 0.979758 0.9816843 0.9834273 0.9850044 0.9864314 0.9877226 0.9888909 0.9899481 0.9909046 0.9917702 0.9925533 0.993262
2
-x
= x -ye
INPUT: clc; funcprot(0) function ydot=f(x,y) ydot=x^2 - (exp(-x)*y); endfunction x=0:0.1:5; y0=0; x0=0; y = ode(y0,x0,x,f); disp(y); plot2d(x,y)
OUTPUT: 0. 0.0003258 0.0025568 0.0084933 0.0198705 0.0383977 0.0657843 0.1037567 0.15407 0.2185141 0.298918 0.3971507 0.5151208 0.6547756 0.8180982 1.0071048 1.2238418 1.4703819 1.748821 2.0612743 2.4098734 2.7967628 3.2240965 3.6940361 4.208747 4.770397 5.3811537 6.0431829 6.7586466 7.5297022 8.3585007 9.2471864 10.197896 11.212756 12.293888 13.4434 14.663395 15.955964 17.32319 18.767147 20.2899 21.893506 23.580014 25.351466 27.209894 29.157327 31.195785 33.327282 35.553829 37.87743 40.300083
+
+ y = 0
INPUT:
clc; funcprot(0) function dx=f(t,x) dx(1)= x(2); dx(2)=-x(1) - (exp(-t)*x(2)); endfunction t=4:0.1:10; x= ode([0;1],0,t,f); disp(x(1,:)); plot2d(t,x(1,:))
OUTPUT: -0.4896668 -0.5649233 -0.3746671 -0.0090468 0.4024989 0.5696218 0.3912047 -0.0244664 -0.3415106
-0.5168011 -0.5387308 -0.5552487 -0.5661999 -0.5714838 -0.5710549 -0.5531553 -0.5358722 -0.5132496 -0.4855158 -0.4529492 -0.415876 -0.3297341 -0.2815255 -0.2305223 -0.1772333 -0.1221898 -0.0659407 0.0479247 0.1044059 0.1598336 0.2136552 0.2653341 0.3143549 0.3602291 0.4407431 0.4745803 0.503673 0.5277311 0.5465149 0.5598372 0.5675651 0.5659871 0.5566975 0.541846 0.5215812 0.4961056 0.4656738 0.43059 0.3479114 0.3011428 0.251366 0.1990785 0.1448025 0.0890804 0.0324689 -0.0811568 -0.1370358 -0.1915452 -0.2441404 -0.2942959 -0.3853128
+
=- y
INPUT: clc; funcprot(0) function dx=f(t,x) dx(1)= x(2); dx(2)=-x(1) - (2*x(2)); endfunction t=4:0.1:10; x= ode([0;1],0,t,f); disp(x(1,:)); plot2d(t,x(1,:))
OUTPUT: 0.0732626 0.067948 0.0629814 0.0583448 0.0540203 0.0499905 0.0462384 0.0427478 0.0395028 0.0364883 0.0336897 0.0310934 0.0286861 0.0264554 0.0243895 0.0224772 0.020708 0.019072 0.0175598 0.0161627 0.0148725 0.0136815 0.0125825 0.0115687 0.010634 0.0097724 0.0089784 0.0082471 0.0075737 0.0069537 0.0063832 0.0058582 0.0053754 0.0049314 0.0045233 0.0041481 0.0038034 0.0034868 0.0031959 0.0029289 0.0026837 0.0024587 0.0022522 0.0020627 0.0018889 0.0017295 0.0015833 0.0014493 0.0013265 0.0012139 0.0011107 0.0010162 0. 0009296 0.0008502 0.0007776 0.0007111 0.0006502 0.0005945 0.0005434 0.0004967 0.000454
Plotting
1.
dx
INPUT: clc; x0=0; x1=0:0.1:10; X=integrate('exp(x)','x',x0,x1); plot(X)
OUTPUT:
2.
dx
INPUT:
clc; x0=-10; x1=0:0.1:10; X=integrate('exp(-x)','x',x0,x1); plot(X)
OUTPUT:
3.
dx
INPUT:
clc; x0=0; x1=0:0.1:10; X=integrate('x*exp(x)','x',x0,x1); plot(X)
OUTPUT:
4.
dx
INPUT: clc; x0=0; x1=0:0.1:10; X=integrate('x*exp(-x)','x',x0,x1); plot(X)
OUTPUT:
5.
dx
INPUT:
clc; x0=0; x1=0:0.1:10; X=integrate('x*sin(x)','x',x0,x1); plot(X)
OUTPUT:
Evaluate
)dx ,
σ
σ =1
INPUT: clc; x0=0; x1=0:0.1:10; Y=1/((2*%pi*((1)^2))^0.5); X=integrate('exp(-((x-2)^2)/(2*(1)^2))*(x+3)','x',x0,x1); Z=Y*X
=1,0.1,0.01 and show it tend to 5
plot(Z) disp(Z)
OUTPUT: 0. 0.0182073 0.0409417 0.0690185 0.1033159 0.1447587 0.1942961 0.2528741 0.3214026 0.4007184 0.4915459 0.5944557 0.7098257 0.8378046 0.9782813 1.1308627 1.2948615 1.4692954 1.6528991 1.8441486 2.041298 2.2424269 2.4454962 2.6484096 2.8490789 3.0454873 3.2357501 3.4181681 3.5912718 3.7538544 3.9049933 4.0440578 4.1707059 4.2848693 4.3867295 4.4766867 4.555323 4.6233639 4.6816386 4.7310417 4.7724987 4.8069346 4.8352485 4.8582927 4.8768581 4.8916637 4.9033514 4.9124845 4.9195492 4.9249587 4.929059 4.9321355 4.9344205 4.9361006 4.9373234 4.9382045 4.9388329 4.9392765 4.9395866 4.9398012 4.9399481 4.9400478 4.9401146 4.9401591 4.9401883 4.9402073 4.9402196 4.9402274 4.9402324 4.9402355 4.9402374 4.9402386 4.9402393 4.9402397 4.94024 4.9402401 4.9402402 4.9402402 4.9402403 4.9402403 4.9402403 tends to 5
σ =0.1
INPUT: clc; x0=0; x1=0:0.1:10; Y=1/((2*%pi*((.1)^2))^0.5); X=integrate('exp(-((x-2)^2)/(2*(.1)^2))*(x+3)','x',x0,x1); Z=Y*X plot(Z) disp(Z)
OUTPUT: 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 2.605D-15 5.486D-12 4.325D-09 0.0000013 0.000145 0.0063063 0.1083516 0.7690792 2.4601058 4.1825267 4.8808502 4.9928073 4.9998283 4.9999984 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. tends to 5
σ =0.01
INPUT: clc; x0=0; x1=0:0.1:10; Y=1/((2*%pi*((.01)^2))^0.5); X=integrate('exp(-((x-2)^2)/(2*(.01)^2))*(x+3)','x',x0,x1); Z=Y*X plot(Z) disp(Z)
OUTPUT: 0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 0. 0. 0. 0. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 5. 5. 5.
0. 0. 2.4960106 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. 5. tends to 5
Fourier series 1) Program to sum
INPUT:
clc; s=0; for n=1:100000 s=s+(0.2)^n end disp(s,"sum=");
OUTPUT: 0.25
2) Evaluate the Fourier coefficients of a given periodic function(square wave)
f(x)
INPUT:
=
–
clc; clf; funcprot(0); m=input("no. of Fourier coefficients you need") l=1/%pi function z=f1(x) z=-1 endfunction integral1=l*integrate('f1','x',-1*%pi,0) function r=f4(x) r=1 endfunction integral2=l*integrate('f4','x',0,%pi) a0=integral1 + integral2 disp(a0,"first Fourier coefficient (a0)") //////gap............
disp("value of a1,a2,-----an") function w=f2(x) w=-1*cos(i*x) endfunction function s=f5(x) s=1*cos(i*x) endfunction for i=1:m integral3=l*integrate('f2','x',-1*%pi,0) integral4=l*integrate('f5','x',0,%pi) an=integral3 + integral4 disp(an) end /////break....... disp("value of b1,b2,----bn") function v=f3(x) v=-1*sin(i*x) endfunction function t=f6(x) t=1*sin(i*x) endfunction for i=1:m integral5=l*integrate('f3','x',-1*%pi,0) integral6=l*integrate('f6','x',0,%pi) bn=integral5 + integral6 disp(bn) end
OUTPUT: no. of Fourier coefficients you need 5 (entered by user)
first Fourier coefficient (a0) 0. value of a1,a2,-----an 0.
0. 0. 0. 0. value of b1,b2,----bn 1.2732395 7.071D-17 0.4244132 -3.272D-17 0.2546479
Frobenius method and special functions
1.
=
=
INPUT: clc; clf; funcprot(0); n=input("enter the value of n ="); m=input("enter the value of m ="); z=integrate('legendre(n,0,x)*legendre(m,0,x)', 'x',-1,1) disp(z);
OUTPUT: enter the value of n = 4 enter the value of m = 4
0.2222222 enter the value of n = 4 enter the value of m = 3
0.
2. Plot
,
INPUT:
l=nearfloat("pred",1); x=linspace(-1,1,200)'; y=legendre(0:5,0,x); clf() plot2d(x,y',leg="p0@p1@p2@p3@p4@p5@p6") xtitle("first 6 legendre polynomials")
OUTPUT:
INPUT:
clc; funcprot(0); x=linspace(0.01,10,5000)'; y=besselh(0:5,x); plot2d(x,y)
OUTPUT:
3. Show recursion relation I. (n+1) + n INPUT:
= (2n+1)x
clc; funcprot(0); n= input("enter value of n"); for x =-1:0.5:1; p=(n+1)*legendre(n+1,0,x)+n*legendre(n-1,0,x); q=((2*n)+1)*x*legendre(n,0,x);
if p==q then disp("the first recursion relation is proved") else disp("the relation is not proved"); end end
OUTPUT: enter value of n 4
the first recursion relation is proved the first recursion relation is proved the first recursion relation is proved the first recursion relation is proved the first recursion relation is proved
II.
=
+ n
INPUT: clc; funcprot(0); n= input("enter value of n"); for n=1:1:5; x=1; p=(n*(n+1)-n*(n-1))/2; q=n*legendre(n,0,x); if p==q then disp("the second recursion relation is proved") else disp("the relation is not proved"); end end
OUTPUT: enter value of n 3
the second recursion relation is proved the second recursion relation is proved
the second recursion relation is proved the second recursion relation is proved the second recursion relation is proved
Integral transform of
INPUT:
///Fourier transform clc; i=sqrt(-1); funcprot(0); function c=f(x) c=exp((i*(2*i)*x)-x^2); /// =2i endfunction a=-999; b=999; integral=intg(a,b,f); disp(integral,"the value of Fourier integral will be")
OUTPUT: the value of Fourier integral will be
4.8180291
Theoretically the Fourier transform of is = 2i gives the value of the Fourier transform = 4.81802909